Chap. I) PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 41 
and derived as did Descartes the first three pairs of amicable numbers from 
2, 8, 64. We shall see that various later writers attributed the rule to 
Descartes. 
Mersenne^^ again in 1644 gave the above three pairs of amicable num- 
bers, the misprints in both^^^ of the numbers of the third pair being noticed 
at the end of his book, and stated there are others innumerable. 
Mersenne®^ in 1647 gave without citation of his source the rule in the 
form 2-2% 2-2"/i<, where 1 = 3-2'' -1, h = 2t+l, s = ht+h-i-t are primes 
[as in (1)]. 
Frans van Schooten,^^^ the younger, showed how to find amicable 
numbers by indeterminate analysis. Consider the pair 4a:, 4yz [x, y, z odd 
primes]; then 
7+3a: = 4i/0, 7-\-7y+7z+Syz = 4x. 
Eliminating x, we get 2 = 34-16/(2/ — 3). The case ^ = 5 gives 2;= 11, ^' = 71, 
yielding 284, 220. He proved that there are none of the type 2x, 2yz, or 
8x, Syz, and argued that no pair is smaller than 284, 220. For 16.x, 16yz, 
he found 2=15-l-256/(i/ — 15), which for y = 47 yields the second known 
pair. There are none of the type 32a;, S2yz, or type 64a:, Myz. For 128a:, 
1281/2, he got 2= 127-^16384/(1/ -127), which for 2/ = 191 yields the third 
known pair. Finally, he quoted the rule of Descartes. 
W. Leyboum^^ stated in 1667 that ''there is a fine harmony between 
these two numbers 220 and 284, that the aliquot parts of the one do make 
up the other . . . and this harmony is not to be found in many other numbers." 
In 1696, Ozanam''^ gave in great detail the derivation of the three known 
pairs of "amiable" numbers by the rule as stated by Descartes, whose name 
was not cited. Nothing was added in the later editions.'^' ^^ 
Paul Halcke^^" gave Stifel's^^ rule, as expressed by Descartes.^^^ 
E. Stone^^^ quoted Descartes' rule in the incorrect form that 2^''pq and 
3-2"p are amicable if p = 3-2'*— 1 and g = 6-2'*— 1 are primes. 
Leonard Euler^^^ remarked that Descartes and van Schooten found only 
three pairs of amicable numbers, and gave, without details, a fist of 30 pairs, 
all included in the later paper by Euler.^^^ 
G. W. Kraft^^^ considered amicable numbers of the type APQ, AR, 
where P, Q, R are primes not dividing A. Let a be the sum of all the divi- 
sors of A . Then 
R+1 = {P+1){Q+1), {R-{-l)a = APQ-^AR. 
Assuming prime values of P and Q such that the resulting R is prime, he 
sought a number A for which A /a has the derived value. For P = 3, Q = 1 1 , 
"'Not noticed in the correction (left in doubt) in Oeuvres de Fermat, 4, 1912, p. 250 (on pp. 
66-7). One error is noted in Broscius*^, Apologia, 1652, p. 154. 
"'Exercitationum mathematicarum libri quinque, Ludg. Batav., 1657, liber V : sectiones triginta 
miscellaneas, sect. 9, 419-425. Quoted by J. Landen.*^ 
""Deliciae Mathematicae, oder Math. Sinnen-Confect, Hamburg, 1719, 197-9. 
*"New Mathematical Dictionary, 1743 (under amicable) . 
"'De numeria amicabilibus. Nova Acta Eruditorum, Lipsiae, 1747, 267-9; Comm. Arith. Coll., 
II, 1849, 637-8. 
-•»Novi Comm. Ac. Petrop., 2, 1751, ad annum 1749, Mem., 100-18. 
